…to construct the Polygon of Forces, by drawing in succession lines parallel and proportional to the different forces, each line beginning at the extremity of the last. If the forces acting at the point are in equilibrium, the polygon formed in this way will be a closed one.

Applications of such reciprocal force diagrams include the method of graphic statics, and also thrust network analysis (see Block and Ochsendorf 2007) for the form-finding of masonry vaults, in which the planar reciprocal network corresponds to the horizontal equilibrium of the nodes.

Kangaroo can generate structures in static equilibrium in a simple and intuitive way using dynamic relaxation. Once we have this equilibrium structure we can rearrange the force vectors to form such a reciprocal diagram. When the structure is modelled as a mesh I found there is a nice easy way to do this rearrangement also with Kangaroo: Scale the edges of the mesh so that their length corresponds to the force along that edge, then for each face, take the start points of its edges and just pull them together with attraction forces, but keeping the edge lengths and orientations fixed (ie allowing only pure translation). If the forces are balanced (which they will be if the simulation has stopped moving) then everything matches up as neatly closed polygons in the resulting reciprocal diagram. We can also optionally rotate the edges by 90° before pulling them together, which reduces the distance they have to move.

(Note that unlike the thrust network analysis approach which actually uses the reciprocal diagram as part of the process of generating the equilibrium structure, here I am just showing a way to generate such a reciprocal diagram from a structure after we have found its equilibrium. So in this case it is used not as an essential step, but as an alternative visual representation which I think can be quite revealing about the behaviour of the structure.)

In the special case where our structure has been form-found by treating the edges of a mesh as zero-rest-length springs of uniform stiffness, this generation of the reciprocal figure becomes even simpler still. Recall that the force in a spring is proportional to its extension relative to its rest length. If the rest length is zero, then the length is the extension, so the edge lengths already correspond to their forces and there is no need for any scaling step (another way of looking at this is that zero rest length springs of the same stiffness correspond to constant force-density elements).

This lead me to thinking about the equivalent for tensile structures. If we relax the edges of a mesh with a non planar boundary, we can get a tensile network in space. Here because the only forces in the system are in the edges themselves (unlike the compression shell example, where there is also a vertical load vector on each point), we can actually form the reciprocal force diagram in 3d space without having to project to the plane. The reciprocal diagram is then another stable 3d tension structure of zero-rest-length springs.

(if you focus on a single edge, you will see that they do not rotate or change length, but only translate)

The word reciprocal is also used in a quite different context to refer to frame structures of mutually supporting elements, also sometimes called nexorades (which I explored a bit here).

As I was finding these reciprocal force diagrams, I noticed that as the lines were moving from the triangulated tension structure to its reciprocal (in the force diagram sense), the intermediate stages actually bore some resemblance to such reciprocal structures (in the nexorade sense). As shown the video above, by adding another force to keep the lines pulled onto each other while moving between the 2 forms, this can be emphasized even more, and means that the intermediate networks also form surfaces.

This suggested to me a connection to Bonnet rotations and associate, or adjoint families of minimal surfaces – see 1, 2, 3 for some more info on these (this also ties back in with my old interest in complex analysis).

The work of Pinkall and Polthier uses nonconforming triangulations for performing these Bonnet rotations, as well as describing the now widely used cotan weighting scheme for calculating discrete minimal surfaces. Kangaroo does now contain an option for such cotan weighted mesh smoothing, but this was not actually used in generating the mesh in the video above, because of how using uniformly weighted zero length springs simplifies the transition to the reciprocal diagram described earlier. However, because I have performed a remeshing so that the mesh triangles are all similar, the difference between these is greatly reduced.

Open question – what sort of linkages might allow a physical realization of these Bonnet rotations as an actual transformable structure?

Coming back to compression structures for a moment – in the last few years there has been something of a resurgence of interest in this area, and some great work not just from architecture and engineering departments, but also from the geometry processing/graphics side, and I think it is very exciting to see these disciplines coming together more.

I should also mention that all of my examples above also make use of the remeshing tools I have been working on recently, for preparing the meshes before relaxation. Since the last time I wrote about dynamic remeshing here, I have made significant improvements to these tools (making use of the Plankton half-edge mesh library developed by Will Pearson and myself), and am finding many applications for them. Having meshes with well formed triangles of near equal edge lengths comes in very handy for all sorts of physical simulation and optimization (some more videos of this here: 1,2). I’m also now exploring more ways of actually carrying out the remeshing at the same time as the form-finding, and some of the papers above suggest to me some very interesting further possibilities in this direction.

Here is one example of using the dynamic remeshing coupled with surface smoothing for a very interactive approach to tensile form-finding:

…

Well, that turned into a longer post than I expected – lots of exciting stuff going on!

That’s about all from me for now, but I’d welcome any comments and discussion.

For some more on graphic statics see the Active Statics site by Simon Greenwold (who also wrote the particle-spring library used by the Cadenary tool, which was one of the early inspirations for Kangaroo), and also the eQuilibrium site from ETHZ.

For a lot more on things like thrust network analysis, force-density, dynamic relaxation and shells in general, the new book Shell Structures for Architecture: Form Finding and Optimization is now out. I had the pleasure of being one of the reviewers, and am very happy to see this book now in print – such a reference on these topics is very welcome, and I hope it will lead to even more interesting development and application of shells in architecture.

I’ve always aimed to make Kangaroo a specifically architectural physics engine. While it shares many characteristics with similar engines used for other purposes, such as games and animation, it has some features that are uniquely suited to designing buildings.

Form-finding and physics-based-modelling often result in curved shapes, with an elegant and natural appearance which is something I greatly enjoy about these methods.

However, the fact remains that most existing architectural geometry consists of straight lines and right-angles.

So the latest addition I have made to Kangaroo is for interaction between boxes, which remain aligned with the global XYZ directions. As shown in the video above, this can be combined with other forces such as gravity and attraction, causing the boxes to cluster together, and they will always remain orthogonal.

Because of the constraint to the component directions, this actually removes the need for calculating any square roots for distances, which is usually a significant part of the physics computation, so it is simple and fast.

I’ve also added an option to allow the boxes to alter their proportions in order to maintain a fixed volume. So as they get pushed on by neighbouring boxes or other forces they may get squashed or elongated, but they keep their contained volume the same (Movement could also be constrained in the vertical direction, so areas are fixed).

Some people have already used Kangaroo for arranging rooms (for example see Marc Syp’s project Realtime Physics for Space Planning), using systems of weighted springs and colliding spheres to balance complex networks of adjacency requirements between different functional areas of buildings. I’m hoping this new functionality could be useful in taking these ideas further…

In architectural geometry over the last few decades, a common topic of research has been how to build and clad doubly curved surfaces in an efficient way.

While computer aided manufacturing has made it possible to make buildings where every panel has slightly different dimensions, there are still often significant benefits in fabrication and assembly if identical panels can be used.

However, it is a geometric impossibility to smoothly cover a doubly curved surface with many perfectly identical panels, all meeting exactly edge-to-edge. The typical way of dealing with this is to have many variations in the shapes of the panels. Non-uniformity in the connectivity of the mesh (ie. some irregular vertices) can help reduce the magnitude of these variations (see my previous post for more on this), but still leaves many different panel shapes.

An approach I’ve been exploring a bit recently is to allow some gaps to open up between panels in a controlled way, but still keeping vertices connected. By each panel rotating slightly in a particular alternating clockwise/counter-clockwise pattern they are able to expand and contract to allow for the curvature of the surface. To keep a high ratio of panel to gap area, this expansion must be limited, and well positioned irregular vertices (the 4 pointed stars in the image above) can help with this.

This can be done with various tiling types – squares, triangles, and hexagon/triangle combinations. However, to allow this alternating twist pattern, it must be one which can be coloured in a checkerboard fashion, which also means every internal vertex must be surrounded by an even number of faces.

Of course, if we want to make such a structure into a closed skin, then something needs to fill in the gaps, which do all vary in shape, but if these are made from a material which is easier to cut, I think there could still be a significant overall benefit. For example, mass-produced identical triangular or square glass panels, with varying aluminium ‘stars’ or ‘diamonds’ for the gaps.

Designing such panelizations is a good application of a computational physics design tool like Kangaroo. Indeed I was inspired by the sculptures of Ron Resch, who produced some similar forms as an emergent result of physical processes of folding paper.

It is not always easy to predict what surfaces it is possible to panel in such a way, and only by setting up the physical constraints and manipulating the surface do we see how it can deform and get a feel for its possibilities.

At a small scale we can actually do this with the real material, but at a larger scale, if we wanted to pre-calculate the shape and the dimensions of the varying gap pieces beforehand then the digital-physical form-finding is vital.

Amazingly Ron Resch actually even wrote some computer programs back in the 60s to do some fairly similar iterative form-finding and even produce 3d animations. (See in the video linked to earlier from about 28:00 to 38:00)

Such examples of what was possible even with relatively primitive technology are an inspiration to push things further today…

As regular readers of this blog will know, I’m passionate about the use of relaxation and force-based methods for optimizing geometry in a very interactive way.

There is a great variety of form-finding that can be done by assigning physical forces as interactions between sets of particles. However, in my investigations so far, the topology of these interactions has usually remained fixed. So the overall shape changes dramatically, but the number of points -and the underlying network of which point interacts with which- remains the same throughout the simulation.

I have written about topology optimization before, in my post on self organizing structures, but the videos above use a fundamentally different approach than any of the examples shown there, as this new work is based on purely local interactions, which reconfigure their connectivity as the geometry changes.

I did experiment a while back with a different technique for dynamic remeshing based on repelling/colliding particles or spheres (see this video, skip to around 1:27 for the mesh generation), but this was limited by a number of factors – although the spheres only exert force on each other when they are closer than a certain distance, they still need to be constantly checked against all the other spheres to see if they are within this distance. Checking every sphere against every other one means the number of interactions increases as the square of the number of particles, which becomes slow for even modest meshes of a few thousand vertices.

There are various ways of significantly reducing this complexity and cutting down the number of distance calculations to be performed at each iteration (such as partitioning the space into cubic cells, and only checking for interactions between particles in the same or neighbouring cells). However, there still remains the question of how to remove or add spheres if they get too close or too spread out. Then even when a fairly equal distribution of points across the surface has been achieved, turning this into a clean mesh can still be tricky.

Instead, using the mesh connectivity itself from the very beginning as the network of which vertices interact with which, and updating it iteratively and locally based on its changing geometry, solves all of these problems at once. This connectivity update can be done by using repetitions and combinations of just 3 essential moves :

edge flip

edge collapse

edge split

(or alternatively, one can use edge flip, vertex insertion and vertex removal)

vertex insertion/removal

I’m always using a triangulated mesh here. In the 2nd and 3rd videos at the start, the underlying physics is still working with triangles, but I’ve shown the dual of this, which exchanges the roles of faces and vertices, turning a mesh of mostly triangles into one of mostly hexagons:

Notice that the irregular vertices of the triangle mesh (ones which have 5 or 7 connected edges instead of the regular 6) correspond to pentagons and heptagons in the dual. Finding the right number and placement of these irregular vertices is an important part of making a good mesh. There is a close relationship between these irregular vertices and the overall topology and curvature of the mesh. Euler’s polyhedron theorem gives us a precise relationship between the number of faces/edges/vertices, and the genus of the mesh (F-E+V=2-2g). This is linked to Descarte’s polyhedron theorem, which relates the total angle defect at the vertices to the genus, and is itself a discrete version of the continuousGauss-Bonnet theorem which describes a similar relationship for the integral of Gaussian curvature in the smooth case.

Keeping a good quality triangular mesh (close to even sized equilateral elements, with no obtuse, skinny or degenerate ones) can be very useful for many other types of optimization and simulation, as well as an advantage for fabrication if it is to be built as a physical structure.

I’m thinking I could also vary the remeshing rules depending on whether the priority is regularity of geometry, or regularity of connectivity. For example, on a sphere, a geodesic dome has a small number of irregular vertices (this is sometimes referred to as a semi-regular mesh), but a fair amount of variation in edge length, whereas solutions to the Thomson or Tammes problems reduce the variation of distance, but have larger numbers of irregular vertices.

If the triangles all have identical edge lengths, then all of the angle defect is concentrated at the irregular vertices, whereas if the edges can vary slightly, the angle defect can be spread out across the mesh. As mentioned above, by Descarte’s theorem, the total angle defect is fixed, but if the number of vertices is increased it can be divided between more of them. Subdividing each triangle (using a smoothing scheme such as Loop subdivision) does not alter the configuration of irregular vertices, but reduces the angle defect at each vertex (and in the smooth limit it approaches zero as the number of vertices approaches infinity).

The remeshing can also be based on other criteria than just equal triangles. Reducing triangle size in high curvature areas is an obvious one, but I have a hunch there could also be some interesting ways of basing it not just on geometry, but on stresses, and using it for structural optimization.

I find remeshing fascinating because the same geometric rules and operations have relevance across so many different disciplines and at a variety of scales.

For example, in the carbon nanostructures graphene and nanotubes, which are hexagonal arrangements of atoms, there is a much studied crystallographic defect which occurs called the Stone-Wales defect – where instead of all hexagons we get 2 pentagons and 2 heptagons. It has important implications for the material’s mechanical and electrical properties. If we look again at our edge-flip move on a regular triangular mesh, and its effect on the dual, we see that it is exactly the Stone Wales defect!

There are even hypothetical carbon nanostructures (proposed by Mackay and Terrones) in the shape of doubly curved triply periodic minimal surfaces known as Schwarzites.

As the bond lengths between the Carbon atoms are very rigid, the only way these curved structures can be formed is through the variations in mesh connectivity.

Because such complex geometries are only dependent on the way in which many identical elements are connected, rather than variation in the elements themselves, they can be modelled using simple materials (no laser cutting involved – just clever assembly!):

So as well being potentially useful for design, meshes and their properties have profound relevance to our understanding of the nature of space and curvature, because of the way they link the discrete and the continuous.

I’ll be continuing to work on these remeshing tools and their combination with other types of relaxation and optimization. These recent developments are not publicly available for now, but I’ll update here with any news.

also – a general update : Around a year ago I started working full time for Foster+Partners in the Specialist Modelling Group. This has been a really great experience, working on (and applying Kangaroo on) some big and exciting projects, but core Kangaroo development (and posts here) did necessarily slow down a bit during this time. However, I have recently left F+P and things will be changing ! More news to follow…

A soliton is a kind of solitary, stable and localised wave which acts in many ways like a particle. They are useful in describing a diverse range of physical phenomena, and their mathematics is a large and active topic of research.

One way of demonstrating the idea of solitons is the coupled pendulum model: Imagine a series of pendulums attached at even spacings along a horizontal rod, free to rotate only in the plane perpendicular to the rod, and connected to their neighbouring pendulums by torsion springs. If the pendulum at one end is pulled over the top of the rod it introduces a twist into the system. The weight of the pendulums makes them want to hang straight down, and along most of the rod they do, but at this twist they cannot. Because of the pendulums pulling down either side of the twist, it remains localised and cannot spread out. It can move along the rod like a wave, but unlike a normal wave, it cannot simply dissipate and disappear (except by leaving one end of the rod). These twists can be in either direction (sometimes called kink and anti-kink solitons), and interestingly they can pass through each other and emerge unchanged. Curious about this behaviour, I had a go at simulating it in Kangaroo :

Another model for solitons can be made with playing cards :

(I think I first encountered this model in the book ‘The New Ambidextrous Universe’ by Martin Gardner)

I see a link here with the the idea of bistable structures. A normal elastic object has a single unstressed configuration, and if you bend it away from this shape(provided you do not bend it too far) it will always try and spring back to that one rest configuration. A bistable object on the other hand has 2 different states in which it is at rest, and in between a more highly stressed state. An example of a bistable structure that some readers may remember are the ‘snap bracelets‘ popular during the early 90’s.

These were very similar to metal measuring tape – a strip of thin metal with a slight curvature across its short direction when straight along its length. A normal metal tape measure is not actually bistable, because it is still stressed when in its rolled state, but it is much less stressed in the rolled or straight configurations than in the in between state.

This allows a sharp bend to move along the tape while staying localized, a bit like the pendulum soliton.

I’ve recently been exploring the idea of whether it is possible to create something similar, but on a surface, not just a linear element.

Here I am also drawing on the idea of auxetic materials – defined as having a negative Poisson’s ratio. This means that unlike most materials which get thinner in cross section as you stretch them along their length, auxetics surprisingly actually get fatter. This can be achieved by small scale structures within the material behaving as linkages.

Many of the origami textured surfaces I have explored in previous posts actually have auxetic properties. Something I am recently trying to do is combine auxetic and bistable properties in a single sheet material through 3d printing. I’ll update when the test print arrives, but I’m hoping that the transition region between the two different contracted states of the material will behave something like a 2 dimensional version of the unstable region of the tape measure, causing some interesting out of plane buckling.

I’ve barely scratched the surface of the real study of solitons, and what I’ve mentioned here are only really toy models of a highly complex subject. For some more related reading :